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Conradie 59-70 Collect. Math. 44 (1993), 59–70 c 1994 Universitat de Barcelona Generalized precompactness and mixed topologies Jurie Conradie Department of Mathematics, University of Cape Town Rondebosch, 7700 South Africa Abstract The equicontinuous sets of locally convex generalized inductive limit (or mixed) topologies are characterized as generalized precompact sets. Uniformly pre- Lebesgue and Lebesgue topologies in normed Riesz spaces are investigated and it is shown that order precompactness and mixed topologies can be used to great advantage in the study of these topologies. Notions of smallness play an important role in analysis, and one of the best known and most useful of these is precompactness. Many smallness properties appearing in the literature can in fact be thought of as “precompactness-like” conditions. A generalized form of precompactness was introduced in [4] in order to prove an ex- tension of Grothendieck’s precompactness lemma. The aim of this paper is to show that in the context of locally convex spaces there is an intimate relationship between generalized precompactness and generalized inductive limit (or mixed) topologies. Generalized precompactness is defined in the first section, and examples are given. Mixed topologies are introduced in the second section and it is shown that the equicontinuous sets of locally convex mixed topologies can often be characterized as generalized precompact sets. In the third section the special case of order pre- compact sets in normed Riesz spaces and the related mixed topologies are explored in more detail. These topologies turn out to be closely related to the uniformly Lebesgue topologies introduced by Nowak in [11]. 59 60 Conradie 1. Generalized precompactness Let E be a vector space (real or complex). A bornology B on E is a collection of subsets of E which covers E, is closed under finite unions and has the property that if B ∈Band A ⊂ B, then A ∈B. Now let E be a topological vector space. Following [4], we call a subset A of E B-precompact if for every neighborhood U of0inE, there is a B ∈Bsuch that A ⊂ B + U. The reader is referred to [4] for the elementary properties of B-precompact sets. Some examples can also be found there; we give some more. Example 1.1: Let E be a Banach space and B the collection of relatively weakly compact subset of E. It follows from a result of Grothendieck ([5], Chapter 13, Lemma 2) that the B-precompact sets are exactly the relatively weakly compact sets in E. Example 1.2: Let E be a normed space and B be the collection of bounded weakly metrizable subsets of E. It follows from [14], Lemma 1.2 that the B-precompact sets are exactly the bounded weakly metrizable sets. Example 1.3: Let E be a locally convex space. If U is a closed absolutely convex neighborhood of 0 in E, we shall denote by EU the locally convex space obtained by equipping E with the gauge of U as seminorm. Let B be the collection of bounded subsets of E. Then E is quasinormable if for every closed absolutely convex neighbor- hood U of 0 in E, there is another such neighborhood V such that V is B-precompact in EU (cf. [9], 10.5.2). Example 1.4: Let A be an ideal of operators between Banach spaces in the sense of Pietsch ([13]). For a Banach space F let B be the collection of all subsets of B of F such that there is a Banach space G and an S ∈A(G, F ) such that B ⊂ S(BG), where BG denotes the closed unit ball of G. A bounded linear operator T : E → F belongs to the surjective hull of the closure of A if and only if T maps the unit ball of the Banach space E into a B-precompact set. (cf. [10]). A bornology B on a vector space E is a vector bornology if it is closed under sums, scalar multiples and balanced hulls. A subset B0 of a bornology B is a basis for B if for every B ⊂B, there is a B0 ∈B0 such that B ⊂ B0. A vector bornology will be called convex if it has a basis consisting of absolutely convex sets. It is easy to check that if B is a bornology on a topological vector space E, the collection Bp of all B-precompact sets is again a bornology on E.IfB is a vector bornology, so is Bp;ifB is convex, Bp is closed under the formation of convex hulls. Generalized precompactness and mixed topologies 61 2. Mixed Topologies The mixed topologies we consider will be a special case of the generalized inductive limit topologies introduced by Garling [7], and a slight generalization of the mixed topologies of Persson [12]. If E is a vector space, B a convex bornology on E and τ a vector topology on E such that every B ∈Bis τ-bounded, we shall call the triple (E,B,τ)amixed space. If in addition B has a basis consisting of τ-closed sets, the mixed space is called normal. The mixed topology γτ (B) is the finest locally convex topology coinciding with τ on the sets in B. If the bornology B is clear from the context, we shall abbreviate γτ (B)toγτ . We write β for the finest locally convex topology for which every B ∈Bis bounded; when equipped with this topology E is a bornological space. The space of all linear functionals on E which are bounded on the sets in B will be denoted by Eb. This is also the dual (E,β) of E equipped with the topology β. The space Eb will always have the topology τb of uniform convergence on the sets of B. It is easy to check that if τ is locally convex, we have τ ≤ γτ ≤ β and hence b b (E,τ) ⊂ (E,γτ ) ⊂ E . Furthermore, (E,γτ ) is a complete subspace of E ([12], Corollary 1.1, Theorem 2.1). In the case where (E,B,τ) is normal, it follows from Grothendieck’s completeness theorem ([15], Chapter VI, Theorem 2) that (E,γτ ) is the closure of (E,τ) in Eb. It follows from [7], Proposition 1, that if (E,B,τ) is a mixed space, B0 a basis for B and τ locally convex, then a basis for the γτ -neighborhoods of 0 is given by the collection of absolutely convex hulls of the sets ∪{(B ∩ UB): B ∈B0}, where (UB)B∈B0 ranges over families of absolutely convex τ-neighborhoods of 0 in E. This description enables us to generalize a result of Cooper ([Co], Proposition 1.22) char- acterizing the γτ -equicontinuous sets. Theorem 2.1 Let (E,B,τ) be a normal mixed space, with τ locally convex and B0 a basis for B consisting of τ-closed sets. A subset A of (E,γτ ) is γτ -equicontinuous if and only if for every ε>0 and every B ∈B0 there is a τ-equicontinuous set A(ε, B) such that A ⊂ A(ε, B)+εB0, where the polar B0 is taken in Eb. 62 Conradie Proof. We first note the every B ∈B0 is σ(E,(E,τ) )-closed, and hence also σ(E,(E,γτ ) )-closed, by [15], Chapter VI, Theorem 2, Corollary 3. If U is an abso- lutely convex closed τ-neighborhood of 0 in E, it can then be shown as in the proof of [15], Chapter VI, Theorem 2 that (U ∩ B)0 ⊂ U 0 + B0 ⊂ 2(U ∩ B)0. If A is a γτ -equicontinuous set and ε>0, it follows from the characterization of the γτ -neighborhoods of 0 that we can find a family (UB)B∈B0 of absolutely convex 0 closed τ-neighborhoods of 0 such that A ⊂ ε[ac ∪{(B ∩ UB): B ∈B0}] , (where “ac” denotes the absolutely convex hull) and hence for every B ∈B0, ⊂ ∩ 0 ⊂ 0 0 A ε(B UB) εUB + εB . 0 The result then follows from the fact that εUB is τ-equicontinuous. Conversely, suppose A ⊂ (E,γτ ) satisfies the given condition. It follows eas- ily from the definition of γτ that A will be γτ -equicontinuous if (and only if) the restrictions of the functionals in A to B is τ-equicontinuous for every B ∈B0.If ε>0 and B ∈B0, then it follows from the assumption that we can find a closed ⊂ 1 0 0 ⊂ ∩ 0 absolutely convex τ-neighborhood UB of 0 such that A 2 ε[UB +B ] ε[UB B] . Hence |f(x)|≤ε for every f ∈ A, x ∈ UB ∩ B, as required. As was pointed out in [4], the fact that for a bornology B on E, ∪B = E,is needed to show that every precompact set is B-precompact. This still holds even if we only assume that ∪B is dense in E. This slightly generalized version of B- precompactness allows us to restate the above theorem. Corollary 2.2 Let (E,B,τ) be a normal mixed space, E the collection of τ-equicontinuous subsets of (E,τ) and let (E,γτ ) have the topology τb. Then a subset of (E,γτ ) is γτ -equicontinuous if and only if it is E-precompact. It follows from the corollary that B-precompact sets in a duality-setting may well signify the presence of a normal mixed space. We illustrate this using Example 1.1. Let E denote the dual of the Banach space E, B the bornology of norm bounded sets in E and τ the Mackey topology τ(E,E).
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